$\max_{i\in \{1,...,M\}} (a_i+ r_i)$ strictly convex?

Question

For any $r\equiv(r_1,...,r_M)\in \mathbb{R}$, consider the function $$a\equiv(a_1,...,a_M) \in \mathbb{R}^M\mapsto G(a)\equiv \max_{i\in \{1,...,M\}} (a_i+ r_i)$$

Is $G$ strictly convex or convex? Could you help me to show it?

Answer 1

The maximum of convex functions is convex.

See here.

Since $\phi_i(a)=a_i+r_i$ is convex for every $i$, then $G(a)=\max \{\phi_i(a)\} $ is convex.

Consider $M=1$ and $r=0$, since $G(a)=a$, $G$ is not strictly convex.